Formula:KLS:14.06:07

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x ⁢ Q ^ n ⁡ ( x ) ⁢ q = Q ^ n + 1 ⁡ ( x ) ⁢ q + [ 1 - ( A n + C n ) ] ⁢ Q ^ n ⁡ ( x ) ⁢ q + A n - 1 ⁢ C n ⁢ Q ^ n - 1 ⁡ ( x ) ⁢ q 𝑥 q-Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 𝑞 q-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 𝑞 delimited-[] 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right){% q}={\widehat{Q}}_{n+1}\!\left(x\right){q}+\left[1-(A_{n}+C_{n})\right]{% \widehat{Q}}_{n}\!\left(x\right){q}+A_{n-1}C_{n}{\widehat{Q}}_{n-1}\!\left(x% \right){q}}}}

Substitution(s)

C n = - α ⁢ q n - N ⁢ ( 1 - q n ) ⁢ ( 1 - α ⁢ β ⁢ q n + N + 1 ) ⁢ ( 1 - β ⁢ q n ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) subscript 𝐶 𝑛 𝛼 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 𝑛 𝑁 1 1 𝛽 superscript 𝑞 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=-\frac{\alpha q^{n-N}(1-q^{n}% )(1-\alpha\beta q^{n+N+1})(1-\beta q^{n})}{(1-\alpha\beta q^{2n})(1-\alpha% \beta q^{2n+1})}}}} &
A n = ( 1 - q n - N ) ⁢ ( 1 - α ⁢ q n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q n + 1 ) ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 1 ) ⁢ ( 1 - α ⁢ β ⁢ q 2 ⁢ n + 2 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝑁 1 𝛼 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 1 1 𝛼 𝛽 superscript 𝑞 2 𝑛 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n-N})(1-\alpha q^% {n+1})(1-\alpha\beta q^{n+1})}{(1-\alpha\beta q^{2n+1})(1-\alpha\beta q^{2n+2}% )}}}}


Proof

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Symbols List

& : logical and
Q ^ n subscript ^ 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{Q}}_{n}}}}  : monic q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqHahn

Bibliography

Equation in Section 14.6 of KLS.

URL links

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